The present invention relates to a defect diagnosis in a friction system, such as bearings and/or gears in a rotational machine, and to a defect diagnosis in a construction system, such as an unbalance in rotating masses and/or a misalignment where two rotational machines run off an axis thereof.
A sudden and/or unexpected stoppage of manufacturing machinery equipment brings about extremely large economic loss. In facilities utilizing the manufacturing equipment, precautionary maintenance is practiced for preventing such trouble such as a sudden unexpected stoppage. As an example of the method of precautionary maintenance for preventing such trouble, a sound or/and a vibration being generated by the machinery equipment in operation is measured so as to determine the condition thereof. Such a procedure is called a xe2x80x9cstate-base precautionxe2x80x9d. Here, the conventional arts will be explained in terms of the state-base precaution, by taking a vibration measurement as an example.
When making the diagnosis of the presence of troubles by measuring the vibration of machinery equipment, it is decided whether the measured vibration exceeds a reference value or not in the magnitude thereof. Ordinarily, two kinds of reference values are prepared for the decisions. Namely, if the detected value exceeds the reference value being the smaller of the reference values, it is found to be in a domain of caution, therefore observation must be performed frequently, though the equipment operation will be continued. If the detected value exceeds the reference value being the larger of the reference values, it is found to be in a domain of danger, therefore equipment operation must be stopped in order for the equipment to be restored or repaired immediately. When the state of the machinery equipment reaches to the caution domain, a time when it reaches to the danger domain is predicted from a guide such as a chart or a graph indicating a tendency in the past of changes from a normal state into the caution domain, so as to provide for production planning which has the highest economic efficiency and a planning of maintenance, including the restoration or repair.
The equipment of the manufacturing machines located in a factory or works of a company varies depending upon the purposes thereof, i.e., there are used various kinds or sorts of machinery, being different in revolution speed, electric power consumption (torque), and/or loads, including large or small values thereof, or a machine having a large vibration or one having a small vibration.
Each reference value for deciding the presence of the defect or failure is unique and characteristic to each particular machinery equipment, and the reference values are decided on the basis of a large accumulated number of sample data representative of the defect or failure conditions, as well as those representative of the normal conditions. However, for fully obtaining the effect of the state-base precaution, an appropriate reference value for decision is needed.
A significant investment of labor is needed to determine the above reference for decision, since enough sample data may not be obtainable due to a rareness or scarcity of the troubles or accident, or since the kinds or sorts of the machines to be inspected is too large in number. Further, there are many companies which cannot apply the state-base maintenance, because of such reasons as that there is no engineer for maintenance work who has applied knowledge in trouble or defect diagnosis, therefore the sample data cannot be gathered.
Though the state-base maintenance is an economical and superior method of maintenance with reduced costs or expenses, the most suitable possible decision reference or criterion must be decided upon so as to apply the state-base maintenance, as mentioned previously. Many companies are unable to apply the state-base maintenance, since this reference for decision cannot be appropriately decided thereby.
According to the present invention, an object is to provide a diagnosis method wherein attention is paid to a difference of a probability density function of amplitude obtained by normalizing a wave-form of a measured signal, such as the wave-form of vibration which is generated by the machine, from a normal distribution, thereby proposing the decision criterion which can be applied in common to the different facilities of many rotational machines, and also to provide a trouble or defect diagnosis apparatus comprising those functions.
The present invention is based upon a principle that the amplitude probability density function of a measured signal, such as the vibration being generated by an object to be detected, for example, a machine, etc., operating under a normal condition, coincides with the normal distribution, however, it is shifted from the normal distribution when trouble or abnormal condition arises in the machine.
According to the present invention, while no component information relating to the measured signal is utilized, such as the amplitude and the vibration number thereof which is generated by the object such as a machine to be inspected, the normalized amplitude probability density function is decided to coincide with the normal distribution or not, fully independent of specifications such as the revolution number, the electric power consumption, the load and the scale of the structure of the machine.
Namely, according to the present invention, there is provided a defect diagnosis method for an object to be inspected, comprising:
detecting a measured signal being generated by said object to be inspected;
expanding orthogonally an amplitude probability density function of a wave-form of the obtained measured signal in a Gram-Charlier series; and
calculating the Gram-Charlier series so as to make a diagnosis of a defect in the object to be inspected.
Here, the defect in the present invention means indicates the condition where the machine operates differently from operation under the normal condition, but not that it is already in an inoperable condition.
Further, according to the present invention, there is provided a defect diagnosis method for an object to be inspected, comprising:
detecting a measured signal being generated by said object to be inspected;
expanding orthogonally an amplitude probability density function of a wave-form of the obtained measured signal in a Gram-Charlier series; and
Further, according to the present invention, there is provided a defect diagnosis method for an object to be inspected, comprising:
detecting a measured signal being generated by said object to be inspected;
expanding a wave-form of the obtained measured signal obtained in a Fourier series to obtain a frequency spectrum;
expanding orthogonally an amplitude probability density function by viewing the obtained frequency spectrum from an axis of an amplitude thereof in a Gram-Charlier series; and
calculating the Gram-Charlier series so as to make diagnosis of a defect in said object to be inspected.
Further, according to the present invention, there is provided a defect diagnosis method for an object to be inspected, comprising:
detecting a measured signal being generated by said object to be inspected;
expanding a wave-form of the obtained measured signal in a Fourier series to obtain a frequency spectrum;
expanding orthogonally an amplitude probability density function by viewing the obtained frequency spectrum from an axis of an amplitude thereof in a Gram-Charlier series; and
calculating a difference from a normal distribution so as to make a diagnosis of a defect in the object to be inspected. In the invention defined in the above, the measured signal may be a vibration.
In the invention defined in the above, the measured signal may be an acoustic, an acoustic emission, fluctuations of current or of effective electric power rather than a vibration.
In the invention defined in the above, the object to be inspected can include a vehicle, an aircraft and a construction other than an ordinary machine.
Furthermore, according to the present invention, there is provided a defect diagnosis apparatus for implementing the defect diagnosis method, comprising a probe with respect to the object to be inspected.
Here, a method with use of an expanding equation of the Gram-Charlier series will be explained in detail below.
The amplitude probability density function of the normal distribution N(xcexc,"sgr"2) can be expressed as follows, assuming that an averaged value is xcexc and a dispersion is "sgr"2.       f    ⁢          (      x      )        =            1              σ        ⁢                              2            ⁢            π                                ⁢          ⅇ              -                                            (                              x                -                μ                            )                        2                                2            ⁢                          σ              2                                          
Here, by making the averaged value xcexc=0 and the dispersion "sgr"2=1, the normalized N(0,1) can be expressed as follows:       f    ⁢          (      x      )        =            1                        2          ⁢          π                      ⁢          ⅇ              -                              x            2                    2                    
where, xcfx86(x) is as follows.       ϕ    ⁢          (      x      )        =      ⅇ          -                        x          2                2            
It is presumed that an arbitrary density function p(x) can be expanded in the following form with use of a normal distribution density function xcfx86(x) and a derived function thereof:                               p          ⁡                      (            x            )                          =                                            c              0                        ⁢                                          ϕ                                  (                  0                  )                                            ⁡                              (                x                )                                              +                                                    c                1                                            1                !                                      ⁢                                          ϕ                                  (                  1                  )                                            ⁡                              (                x                )                                              +                                                    c                2                                            2                !                                      ⁢                                          ϕ                                  (                  2                  )                                            ⁡                              (                x                )                                              +                                                    c                3                                            3                !                                      ⁢                                          ϕ                                  (                  3                  )                                            ⁡                              (                x                )                                              +                                                    c                4                                            4                !                                      ⁢                                          ϕ                                  (                  4                  )                                            ⁡                              (                x                )                                              +          …                                    (        1        )            
where the xcfx86(n)(x) is expressed by a polynomial of Hermite as below.
xcfx86(n)(x)=(xe2x88x921)nHn(x)xcfx86(x)xe2x80x83xe2x80x83(2)
And the equation (1) is derived as follows.                               p          ⁡                      (            x            )                          =                                            c              0                        ⁢                                          H                0                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                              -                                                    c                1                                            1                !                                      ⁢                                          H                1                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                              +                                                    c                2                                            2                !                                      ⁢                                          H                2                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                              -                                                    c                3                                            3                !                                      ⁢                                          H                3                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                              +                                                    c                4                                            4                !                                      ⁢                                          H                4                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                              -          …          +                                                    c                n                                            n                !                                      ⁢                                          H                n                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                              -          …                                    (        3        )            
Then, by differentiating the normal distribution density function xcfx86(x), the following xcfx86(n)(x) is obtained:
xcfx86(1)("khgr")=xe2x88x92xexe2x88x92x22=xe2x88x92xxc2x7xcfx86(x)
xcfx86(2)(x)=xe2x88x92xcfx86(x)xe2x88x92xxc2x7xcfx86(1)(x)=xe2x88x92xcfx86(x)+x2xc2x7xcfx86(x)=(x2xe2x88x921)xc2x7xcfx86(x)
xe2x80x83xcfx86(3)(x)=2xxc2x7xcfx86(x)+(x2xe2x88x921)xc2x7xcfx86(1)(x)=2xxc2x7xcfx86(x)xe2x88x92(x2xe2x88x921)xc2x7xxc2x7xcfx86(x)=xe2x88x92(x3xe2x88x923x)xc2x7xcfx86(x)
xcfx86(4)(x)=(xe2x88x923x2+3)xc2x7xcfx86(x)xe2x88x92(x3xe2x88x923x)xc2x7xcfx86(1)(x)=(xe2x88x923x2+3)xc2x7xcfx86(x)+(x3xe2x88x923x)xc2x7xxc2x7xcfx86(x)=(x4xe2x88x926x2+3)xc2x7xcfx86(x)
xcfx86(5)(x)=(4x3xe2x88x9212x)xc2x7xcfx86(x)+(x4xe2x88x926x2+3)xc2x7xcfx86(1)(x)=(4x3xe2x88x9212x)xc2x7xcfx86(x)xe2x88x92(x4xe2x88x926x2+3)xc2x7xxc2x7xcfx86(x)=xe2x88x92(x5xe2x88x9210x3+15x)xc2x7xcfx86(x)
xcfx86(6)(x)=(xe2x88x925x4+30x2xe2x88x9215)xc2x7xcfx86(x)xe2x88x92
(x5xe2x88x9210x3+15x)xc2x7xcfx86(1)(x)=
(xe2x88x925x4+30x2xe2x88x9215)xc2x7xcfx86(x)+
(x5xe2x88x9210x315x)xc2x7xxc2x7xcfx86(x)=
(x6xe2x88x9215x4+45x2xe2x88x9215)xc2x7xcfx86(x)
From those differentiating values and the above equation (2), the following can be obtained:
H0(x)=1
H1(x)=x
H2(x)=x2xe2x88x921
H3(x)=x3xe2x88x923x
H4(x)=x4xe2x88x926x2+3
H5(x)=x5xe2x88x9210x3+15x
H6(x)=x6xe2x88x9215x4+45x2xe2x88x9215
Further, the polynomial of Hermite has an orthogonality as below:                                           ∫                          -              ∞                        ∞                    ⁢                                                    H                m                            ⁡                              (                x                )                                      ⁢                                          H                n                            ⁡                              (                x                )                                      ⁢                          ϕ              ⁡                              (                x                )                                                    =                                            δ              mn                        ⁡                          (              x              )                                =                      xe2x80x83                    ⁢                                    m              !                        ⁢                          xe2x80x83                        ⁢                          (                              m                =                n                            )                                                              =                  xe2x80x83                ⁢                  0          ⁢                      xe2x80x83                    ⁢                      (                          m              ≠              n                        )                              
An integration is executed by multiplying Hn(x) at the both sides of the equation (3):             ∫              -        ∞            ∞        ⁢                            H          n                ⁢                  (          x          )                    ⁢              p        ⁢                  (          x          )                    ⁢              ⅆ        x              =                    c        0            ⁢                        ∫                      -            ∞                    ∞                ⁢                                            H              n                        ⁢                          (              x              )                                ⁢                                    H              0                        ⁢                          (              x              )                                ⁢                      ϕ            ⁢                          (              x              )                                            -                            c          1                          1          !                    ⁢                        ∫                      -            ∞                    ∞                ⁢                                            H              n                        ⁢                          (              x              )                                ⁢                                    H              1                        ⁢                          (              x              )                                ⁢                      ϕ            ⁢                          (              x              )                                            +                            c          2                          2          !                    ⁢                        ∫                      -            ∞                    ∞                ⁢                                            H              n                        ⁢                          (              x              )                                ⁢                                    H              2                        ⁢                          (              x              )                                ⁢                      ϕ            ⁢                          (              x              )                                            -                            c          3                          3          !                    ⁢                        ∫                      -            ∞                    ∞                ⁢                                            H              n                        ⁢                          (              x              )                                ⁢                                    H              3                        ⁢                          (              x              )                                ⁢                      ϕ            ⁢                          (              x              )                                            +                            c          4                          4          !                    ⁢                        ∫                      -            ∞                    ∞                ⁢                                            H              n                        ⁢                          (              x              )                                ⁢                                    H              4                        ⁢                          (              x              )                                ⁢                      ϕ            ⁢                          (              x              )                                            -    …    +                            c          n                          n          !                    ⁢                        ∫                      -            ∞                    ∞                ⁢                                            H              n                        ⁢                          (              x              )                                ⁢                                    H              n                        ⁢                          (              x              )                                ⁢                      ϕ            ⁢                          (              x              )                                            -    …  
Since the Hermite polynomial has the orthogonality, only the clauses having the same order, and derived is the following:                                                                         (                                  -                  1                                )                            n                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                                    H                    n                                    ⁡                                      (                    x                    )                                                  ⁢                                  p                  ⁡                                      (                    x                    )                                                  ⁢                                  ⅆ                  x                                                              =                      c            n                          ⁢                  
                ⁢                                                                              c                  0                                =                                  xe2x80x83                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                            p                      ⁡                                              (                        x                        )                                                              ⁢                                          ⅆ                      x                                                                                                                                                                c                  1                                =                                  xe2x80x83                                ⁢                                  -                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                  x                        ·                                                  p                          ⁡                                                      (                            x                            )                                                                                              ⁢                                              ⅆ                        x                                                                                                                                                                                      c                  2                                =                                  xe2x80x83                                ⁢                                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                  (                                                                              x                            2                                                    -                          1                                                )                                            ⁢                                              p                        ⁡                                                  (                          x                          )                                                                    ⁢                                              ⅆ                        x                                                                              =                                                                                    ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        x                          2                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                      -                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                                                                                                                                              c                  3                                =                                  xe2x80x83                                ⁢                                                      -                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        (                                                                                    x                              3                                                        -                                                          3                              ⁢                              x                                                                                )                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                                      =                                                            -                                                                        ∫                                                      -                            ∞                                                    ∞                                                ⁢                                                                              x                            3                                                    ⁢                                                      p                            ⁡                                                          (                              x                              )                                                                                ⁢                                                      ⅆ                            x                                                                                                                +                                          3                      ⁢                                                                        ∫                                                      -                            ∞                                                    ∞                                                ⁢                                                                              x                            ·                                                          p                              ⁡                                                              (                                x                                )                                                                                                              ⁢                                                      ⅆ                            x                                                                                                                                                                                                                                                            c                    4                                    =                                      xe2x80x83                                    ⁢                                                                                    ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        (                                                                                    x                              4                                                        -                                                          6                              ⁢                                                              x                                2                                                                                      +                            3                                                    )                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                      =                                                                                            ∫                                                      -                            ∞                                                    ∞                                                ⁢                                                                              x                            4                                                    ⁢                                                      p                            ⁡                                                          (                              x                              )                                                                                ⁢                                                      ⅆ                            x                                                                                              -                                              6                        ⁢                                                                              ∫                                                          -                              ∞                                                        ∞                                                    ⁢                                                                                    x                              2                                                        ⁢                                                          p                              ⁡                                                              (                                x                                )                                                                                      ⁢                                                          ⅆ                              x                                                                                                                          +                                                                      ⁢                                  xe2x80x83                                                                                                                          xe2x80x83                                ⁢                                  3                  ⁢                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                  p                        ⁡                                                  (                          x                          )                                                                    ⁢                                              ⅆ                        x                                                                                                                                                                                      c                  5                                =                                  xe2x80x83                                ⁢                                                      -                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        (                                                                                    x                              5                                                        -                                                          10                              ⁢                                                              x                                3                                                                                      +                                                          15                              ⁢                              x                                                                                )                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                                      =                                                            -                                                                        ∫                                                      -                            ∞                                                    ∞                                                ⁢                                                                              x                            5                                                    ⁢                                                      p                            ⁡                                                          (                              x                              )                                                                                ⁢                                                      ⅆ                            x                                                                                                                +                                                                                                                                            xe2x80x83                                ⁢                                                      10                    ⁢                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        x                          3                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                                      -                                      15                    ⁢                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        x                          ·                                                      p                            ⁡                                                          (                              x                              )                                                                                                      ⁢                                                  ⅆ                          x                                                                                                                                                                                                              c                  6                                =                                  xe2x80x83                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                            (                                                                        x                          6                                                -                                                  15                          ⁢                                                      x                            4                                                                          +                                                  45                          ⁢                                                      x                            2                                                                          -                        15                                            )                                        ⁢                                          p                      ⁡                                              (                        x                        )                                                              ⁢                                          ⅆ                      x                                                                                                                                              =                                  xe2x80x83                                ⁢                                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                  x                        6                                            ⁢                                              p                        ⁡                                                  (                          x                          )                                                                    ⁢                                              ⅆ                        x                                                                              -                                      15                    ⁢                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        x                          4                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                                      +                                      45                    ⁢                                                                  ∫                                                  -                          ∞                                                ∞                                            ⁢                                                                        x                          2                                                ⁢                                                  p                          ⁡                                                      (                            x                            )                                                                          ⁢                                                  ⅆ                          x                                                                                                      -                                                                                                                          xe2x80x83                                ⁢                                  15                  ⁢                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                  p                        ⁡                                                  (                          x                          )                                                                    ⁢                                              ⅆ                        x                                                                                                                                                    (        4        )            
Here, it is assumed that n number of time series data xcexi are collected from the vibration data which the machine generates.
Then, the average value xcexc is obtained as follows, and the total data is shifted by the average value (xcexi=xcexixe2x88x92xcexc).   μ  =                    ∑                  i          =          1                n            ⁢              λ        i              n  
Next, an effective value "sgr" is obtained as is expressed below, so as to normalize the total data (xcexi=xcexi/"sgr").   σ  =                              ∑                      i            =            1                    n                ⁢                  λ          i          2                    n      
The following summations are obtained from the time series data which are normalized:             s      0        =    n    ,      xe2x80x83    ⁢            s      1        =                  ∑                  i          =          1                n            ⁢              λ        i              ,      xe2x80x83    ⁢            s      2        =                  ∑                  i          =          1                n            ⁢              λ        i        2              ,      xe2x80x83    ⁢            s      3        =                  ∑                  i          =          1                n            ⁢              λ        i        3              ,      
    ⁢            s      4        =                  ∑                  i          =          1                n            ⁢              λ        i        4              ,      xe2x80x83    ⁢            s      5        =                  ∑                  i          =          1                n            ⁢              λ        i        5              ,      xe2x80x83    ⁢            s      6        =                  ∑                  i          =          1                n            ⁢              λ        i        6              ,  …
Then, those summations are divided by the number of the data as follows:                     s        0            =                        n          /          n                =        1              ,          xe2x80x83        ⁢                  s        1            =                                                  ∑                              i                =                1                            n                        ⁢                          λ              i                                n                =        0              ,          xe2x80x83        ⁢                  s        2            =                                                  ∑                              i                =                1                            n                        ⁢                          λ              i              2                                n                =        1              ,          xe2x80x83        ⁢                  s        3            =                                    ∑                          i              =              1                        n                    ⁢                      λ            i            3                          n                                s        4            =                                    ∑                          i              =              1                        n                    ⁢                      λ            i            4                          n              ,          xe2x80x83        ⁢                  s        5            =                                    ∑                          i              =              1                        n                    ⁢                      λ            i            5                          n              ,          xe2x80x83        ⁢                  s        6            =                                    ∑                          i              =              1                        n                    ⁢                      λ            i            6                          n              ,    …  
By inserting them into those coefficient cj, then the following is obtained:
c0=s0=1
c1=xe2x88x92s1=0
c2=s2xe2x88x92s0=0
c3=xe2x88x92s3+3s1=xe2x88x92s3
c4=s4xe2x88x926s2+3s0=s4xe2x88x923
c5=xe2x88x92s5+10s3xe2x88x9215s1=xe2x88x92s5+10s3
c6=s6xe2x88x9215s4+45s2xe2x88x9215s0=s6xe2x88x9215s4+30
By the normalization as in the above, c0=1 and c1=c2=0, therefore a compensated clause starts with the third clause.
The coefficient cj can be expressed as below.
c3=xe2x88x92s3
c4=s4xe2x88x923
c5=xe2x88x92s5+10s3
c6=s6xe2x88x9215s4+30
The series being obtained by the expansion in this manner is called the Gram-Charlier series. The Gram-Charlier series contain therein elements of the defect or trouble and deterioration. However, mathematically, as the Gram-Charlier series are in orthogonal relationships with each other, the elements of the defects and deterioration contained therein can be considered to be totally independent of each other.
When the amplitude probability density function is the normal distribution, all of the Gram-Charlier series become zero. When the condition comes into the defect or trouble and shifts from the normal distribution, each absolute value of these numeral (series) values becomes large. Accordingly, by determining the criterion or reference value of the Gram-Charlier series, it is possible to make diagnosis of the defects or troubles.
From the equation (3), the arbitrary density function p(x) becomes as follows:                               p          ⁡                      (            x            )                          =                              (                          1              +                                                                    c                    3                                                        3                    !                                                  ⁢                                  (                                                            x                      3                                        -                                          3                      ⁢                      x                                                        )                                            +                                                                    c                    4                                                        4                    !                                                  ⁢                                  (                                                            x                      4                                        -                                          6                      ⁢                                              x                        2                                                              +                    3                                    )                                            +                                                                    c                    5                                                        5                    !                                                  ⁢                                  (                                                            x                      5                                        -                                          10                      ⁢                                              x                        3                                                              +                                          15                      ⁢                      x                                                        )                                            +                                                                    c                    6                                                        6                    !                                                  ⁢                                  (                                                            x                      6                                        -                                          15                      ⁢                                              x                        4                                                              +                                          45                      ⁢                                              x                        2                                                              -                    15                                    )                                            +              …                        ⁢                          xe2x80x83                        }                    ⁢                      ϕ            ⁡                          (              x              )                                                          (        5        )            
This is called a Gram-Charlier distribution function. This Gram-Charlier distribution function is the function which is the most analogous or approximated to the amplitude probability density function which is actually measured.
The arbitrary density function can be expressed by a sum of the normal distribution density function xcfx86(x) and a differentiation function r(x), as shown in the following equation:
p(x)=xcfx86(x)+r(x)
where the difference function r(x) becomes as below from the equation (5):                               r          ⁡                      (            x            )                          =                                            p              ⁡                              (                x                )                                      -                          ϕ              ⁡                              (                x                )                                              =                                    (                                                                                          c                      3                                                              3                      !                                                        ⁢                                      (                                                                  x                        3                                            -                                              3                        ⁢                        x                                                              )                                                  +                                                                            c                      4                                                              4                      !                                                        ⁢                                      (                                                                  x                        4                                            -                                              6                        ⁢                                                  x                          2                                                                    +                      3                                        )                                                  +                                                                            c                      5                                                              5                      !                                                        ⁢                                      (                                                                  x                        5                                            -                                              10                        ⁢                                                  x                          3                                                                    +                                              15                        ⁢                        x                                                              )                                                  +                                                                            c                      6                                                              6                      !                                                        ⁢                                      (                                                                  x                        6                                            -                                              15                        ⁢                                                  x                          4                                                                    +                                              45                        ⁢                                                  x                          2                                                                    -                      15                                        )                                                  +                …                            ⁢                              xe2x80x83                            )                        ⁢                          ϕ              ⁡                              (                x                )                                                                        (        6        )            
Then, the next equation is defined:                     δ        =                              ∫                          -              6                        6                    ⁢                                                    (                                  r                  ⁡                                      (                    x                    )                                                  )                            2                        ⁢                          ⅆ              x                                                          (        7        )            
where a square integration value of the difference function r(x) from 6 to xe2x88x926 is the difference xcex4.
The difference xcex4 is zero when the amplitude probability density function is coincident with the normal distribution, while the numeral value comes to be large when being in the condition of the defect or trouble and shifted from the normal distribution. Accordingly, by determining the criterion or the reference value of the difference xcex4, it is possible to make the diagnosis of the defect or trouble.